Rosenzweig-MacArthur Model. Considering the Function that Protects a Fixed. Amount of Prey for Population Dynamics

Transcription

1 Contemporary Engineering Sciences, Vol. 11, 18, no. 4, HIKAI Ltd, osenzweig-macarthur Model Considering the Function that Protects a Fixed Amount of Prey for Population Dynamics E.J. Cañate-Gonzalez 1, W. Fong-Silva 1, C.A. Severiche-Sierra, Y.A. Marrugo-Ligardo and J. Jaimes-Morales 1 Universidad de Cartagena, GIMIFEC esearch Group Cartagena de indias, Colombia Universidad de Cartagena, MAAS esearch Group Cartagena de indias, Colombia Copyright 18 E.J. Cañate-Gonzalez et al. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Populations can grow, disappear or remain in time and this depends on their birth rates, mortality, immigration and emigration. For a long time, species have disappeared and others are at the same risk if protective policies are not implemented to guarantee their survival. It is necessary to consider mathematical models that help conserve the environment. The analysis of the osenzweig - MacArthur model was performed considering the function that protects a fixed amount of prey. For the analytical study, the qualitative theory of the systems of differential equations is used. Considering the use of shelters by dams with the functions proposed by Maynard-Smith. It is shown that the refuge influences the stability of the only equilibrium point inside the first quadrant and produces important effects in the dynamics of the prey-prey model. Keywords: Population dynamics, Predator prey models, efuge function Introduction In population dynamics, the models proposed in the literature for different inter-

2 1196 E.J. Cañate-Gonzalez et al. actions between species, consider assumptions with the aim of simplifying their mathematical descriptions [1]. Some of them are: - Homogeneity of the populations: Sex, age and size differences are not considered in the populations. - Homogeneity of the Medium: The physical and biological characteristics of the medium remain constant and the medium is not affected by external conditions. - Uniform spatial distribution. - Constant growth rates - Encounters between predatory species and equally probable prey. - Population sizes dependent exclusively on time - They do not consider behaviors of the species of physiological, morphological, social type, etc. -The predatory species feeds exclusively on the prey species, while it feeds on a resource that is found in the habitat in large quantities, which only intervenes passively. Physiological, morphological, and social behaviors are not considered. The behavior of the species can be affected by ecological variables such as refuge availability, formation of defense groups, emergence of antidepredatory strategies such as []: - Physiological: Emission of chemical substances or pheromones, etc. - Morphological: Patterns of coloration, mimicry with the environment, adaptability of some parts of the body, etc. - Imitation of codes: Emission of sounds. - Adaptation of habitat: Uses of shelters, in nature many dams respond to attacks by predators looking for space shelters such as a shell or a burrow. The qualitative analysis of the two-dimensional systems of autonomous differential equations of osenzweig-macarthur, representing a predator-prey interaction, is carried out [3]. Considering the use of shelters by dams with the functions proposed by Maynard-Smith [4]. epresented by the following system of differential equations: ( ) d q( 1 β ) Y dt = r K ( 1 β ) a β + µ dy p( 1 β ) dt = b ( 1 ) cy β + a ( 1 ) = ( ) γ µ ( 1 ) q( α ) α + p( α ) ( α + cy ) d Y dt r = K a = dy dt = b a where: It is the density of the prey species It is the density of the predatory species

3 osenzweig-macarthur model considering the function 1197 It is the per capita growth rate of the prey species q: It is the rate of encounter between species a: It is the amount of prey needed to obtain half of It is the rate of conversion of dams into new predators It is the natural death rate of predators. K: The load capacity or support of the environment. p: efficiency with which predators convert consumed prey into new predators. osenzweig-macarthur model If and Y represent the densities of prey and predators, respectively, the classic osenzweig-macarthur model has the following form [3]: d qy = r (1 ) dt K + a dy p = b ( cy ) dt + a (1.5) Predator prey models with refuge for prey If, and are continuous variables that represent the densities of the prey, predator and prey species that take refuge, respectively, then the number of prey that interact with the predators is. The effect of the refuge is considered in the functional response of the predator in equation (1.1). The mathematical expression that represents the biological prey-predator interaction system considering the refuge for prey is: d = r ( ) f( r, YY ) dt dy = χ f( r, Y) δ( YY ) dt (1.6) Maynard-Smith (1974) [4], proposed two functions for the prisoners who take refuge and are: 1. The one that protects a constant fraction of the dams, whose equation is: and its graph is: r = β r e =

4 1198 E.J. Cañate-Gonzalez et al.. The one that protects a fixed amount of prey whose equation and graph are: r = β The functions, proposed by Maynard-Smith to model the number of prey that take refuge, have the following objections: The first says that the fraction of hidden prey is a growing linear function, this implies that the availability of refuge must be greater the more large is the population size. The second indicates that the fraction of prey is fixed regardless of the availability of the refuge. "The occurrence of a constant number or a constant proportion of the prey in refuge seems to be very unlikely in nature" (Sih, 1987a) [5]. Almanza et al. (15) [6] proposes the monotonous growing and bounded function to express the dams that take refuge, where is the maximum capacity of the refuge and is the amount of dams needed to occupy half of the maximum refuge capacity, this function presents a solution alternative to the previous objections and also satisfies that: r = α + β

5 osenzweig-macarthur model considering the function 1199 Analysis of the Model Analyze the effect of adding a shelter that protects a fixed amount of prey, r = α in osenzweig-macarthur prey-predator model, which is expressed by the system of autonomous differential equations. ( 1 ) d q( α ) Y dt = r K α + a =.... (3. 1) dy p( ) dt = b( α + a cy ) α µ α Where µ α + 8 = ( abck,,,, pqr,,, ) is the vector of biological parameters. Condition when α = a When the (3.1) is expressed by: ( 1 ) = r =...(3. ) p( ) ( ) d q( α ) Y α dt K µ dy α dt = b cy When realizing a reparameterization of the coordinates, re-escalating the time, with the methodology proposed by Sáez and González (1999) [7], the dimensionless system is obtained. η dn dτ = (1 NN ) ( N P ) =...(3.3) dp dτ = BN ( (1 C) P ) bp α c Where; B = ; = ; C = and the diffeomorphism between the systems r K p of differential equations that makes, the systems are topologically equivalent [8], [9]. It is expressed by: rk rτ ϕ( N, P, τ ) = ( KN, P, ) = (, Y, t) r K with det Dϕ( N, P, τ ) = qn >, The new vector field in the new coordinates es = ϕ α [1, 11]. η µ The region of Invariance is ( ) q N + { N, P ( ) / N 1, P } Ω=

6 1 E.J. Cañate-Gonzalez et al. The Jacobian matrix associated with the system η is given by: 3 ( ) ( ; ) N + N P N J η N P = B(1 CP ) BN ( (1 C) ) The equilibrium points with biological sense of the system η are: i). ( N, P ) = (,) It is not considered for not belonging to the region of invariance ii). ( N, P ) = (1, ) (1 C ) iii). ( N, P ) = (, ) where 1 C > y 1 C > 1 C C(1 C) (1 C ) Point ( N, P ) (, = ) collapse with ( N, P ) = (1, ) when 1 C C(1 C) 1 C 3 Theorem 4. For all η = ( BC,, ) + the singularity ( N = 1, P = ) is: Hyperbolic chair, yes 1 C > It is a locally stable attractor focus, if1 C < Demonstration The Jacobian matrix evaluated at the critical point ( N = 1, P = ), is given by J η 1 (1 ) (1, ) = B((1 C) ) The eigenvalues are: λ 1 = 1< y λ = B((1 C) ), the sign λ of depends on (1 C) The following theorem shows that the equilibrium point inside the first quadrant can change its stability. Theorem 5. For all = (,, ) the singularity 3 η ABC + It is globally asymptotically stable if S < (1 C ) ( N, P ) = (, ) 1 C C(1 C) It is an unstable balance point and around there is a limit cycle, if S > Where S= (1 3 C ) (1 C)(1 C)

7 osenzweig-macarthur model considering the function 11 Demonstration: (1 C ) The Jacobian matrix evaluated at the critical point ( N, P ) = (, ) 1 C C(1 C) is S C C(1 C) 1 C Jη ( N, P ) = B(1 C ) (1 CC ) B (1 C ) It is then that DetJη ( N, P ) = > the eigenvalues depend on (1 C) S the TrazaJη ( N, P ) = Yes S >, the point ( N, P ) is unstable and C(1 C) by the Poincare-Bendixon theorem there is a limit cycle around the point. Yes N, P is locally asymptotically stable. S <, the point ( ) Condition when α a When realizing a reparameterization of the coordinates, re-escalating the time, with the methodology proposed by Sáez and González (1999) [7], of the system (3.1) the dimensionless system is obtained η dn dτ = (1 N) N( N + A) ( N ) P =...(3.4) dp dτ = B( N C( N + A)) P a bp α c Where; A = ; B = ; = ; C = and the diffeomorphism between the K r K p systems of differential equations that makes the systems are topologically equivalent [8], [9]. It is expressed by: rk rτ ϕ( N, P, τ) = ( KN, qp, N + A) = (, Y, t) r K with det Dϕ( N, P, τ) = q( N + A) >, The new vector field in the new coordinates is = ϕ α [1]. η µ The balance points of the system (3.4) are: ( N, P ) = (,) ; ( N, P ) = (1, )

8 1 E.J. Cañate-Gonzalez et al. ( N, P ) = ( A,) For A> AC (1 C) ( AC (1 C) ) H ( N, P ) = ( +, + ) (1 C) C(1 C) Where H = (1 C)(1 L) AC > y 1 C >, and what collapses with, (1, ), when H. It is interesting to see that for = α a the field vector (3.1) it is not defined, but it is possible to make a continuous extension to the point ( α a,) The region of Invariance is already ( ) + { N, P ( ) / N 1, P } Ω= as dn 1. Yes N =, you have to and A (1 ) dτ = > dp and ABC dτ = <. dn. For N = 1, you get and, where (1 P ) dτ = < dp and BHP dτ =, where H = (1 C)(1 ) + AC ; i.e. i.e. independent of the parameters, any path crosses the line N = 1 goes to the interior of Ω. dn 3. For N = dτ = dp and ABCP dτ = < therefore it is a region of variations The Jacobian matrix associated with the system (3.4) is given by: 3 ( ) ( ) ( ) ( ; ) N N A + N A P N J η N P = B(1 CP ) BN ( (1 C) CA ( )) Analysis of Balance points For α > a, the points: ( N, P ) = (,) For, the points: It is not considered for not belonging to the region of invariance and would have a negative time contraction. ( N, P ) = ( A,) It is not considered why time would dilate indefinitely For α < a, the points: ( N, P ) = ( A,) It is not considered why time would dilate indefinitely

9 osenzweig-macarthur model considering the function 13 Theorem 6. For < A ( α < a ) the singularity ( N =, P = ) is Hyperbolic Chair Demonstration: The Jacobian matrix evaluated at the critical point is given by ( N =, P = ), is given by (;) A J η = B( + C( A )) The eigenvalues are: λ 1 = A > and λ = B( + C( A )) < Theorem 7. If H = (1 C)(1 L) AC > the singularity ( N, P ) = (1, ) It is an unstable chair point. If H = (1 C)(1 L) AC > the singularity ( N, P ) = (1, ) It is a locally stable point. Demostration: The Jacobian matrix evaluated at the critical point is given by ( N = 1, P = ), is given by (1 + ) (1 ) J (1; ) A η = BH The eigenvalues are: λ 1 = (1 + A ) <, so much for < A or for > A because if it were not like that > 1+ A, contradicting that < 1. λ = BH whose sign depends on the sign of H Theorem 8. Given 3 S = (1 C) (1 C)( AC + (1 C) ) + AC (1 C AC A) that the point AC (1 C) ( AC (1 C) ) H has ( N, P ) = ( +, + ) (1 C) C(1 C) i). It is an unstable equilibrium point with a limit cycle around it, if S > ii). It is a locally stable attractor focus S < Demonstration: The Jacobian matrix evaluated in ( N, P )

10 14 E.J. Cañate-Gonzalez et al. AC T( AC,, ) 1 C Jη ( N, P ) = B(1 C)( + AC) H (1 CC ) where T( AC,, ) = C(1 C) C AC AC C C C C A C C A AC AC AB(1 C)( + AC) H whose sign depends on the sign of DetJη ( N, P ) = > (1 C) S and TrazaJη ( N, P ) = T ( A, C, ) = C(1 C) Yes S >, the point ( N, P ) is unstable and by the Poincare-Bendixon theorem there is a limit cycle around the point. S <,the point ( N, P ) Yes, Conclusion is locally asymptotically stable. From the results presented, from their discussion, from the demonstrations and from the background of the literature exposed through the article, the following relevant conclusion can be obtained: a trend of limit cycles around the points of equilibrium interior of the first quadrant. It is important to note that parameter B does not influence the equilibria and properties of the speed of the model. eferences [1] A. Wasike, S. Bonga ng a, G. Lawi, M. Nyukuri, A Predator-Prey Model with a Time Lag in the Migration, Applied Mathematical Sciences, 8 (14), [] K. Pusawidjayanti, A. Suryanto and. B. E. Wibowo, Dynamics of a Predator-Prey Model Incorporating Prey efuge, Predator Infection and Harvesting, Applied Mathematical Sciences, 9 (15), [3] M. osenzweig and. MacArthur, Graphical representation and stability conditions of predator-prey interactions, Am. Naturalist, 97 (1963),

11 osenzweig-macarthur model considering the function 15 [4] J. Maynard-Smith, Models in Ecology, Cambridge, Cambridge University Press, [5] A. Sih, Antipredator responses and the perception of danger by mosquito larvae, Ecology, 67 (1986), [6] E. Almanza-Vasquez, uben-dario Ortiz-Ortiz, Ana-Magnolia Marin- amirez, Bifurcations in the Dynamics of osenzweig-macarthur Predator-Prey Model Considering Saturated efuge for the Preys, Applied Mathematical Sciences, 9 (15), [7] E. Sáez and E. González-Olivares, Dynamic of a Predator- Prey model, SIAM Journal on Applied Mathematics, 59 (1999), [8] A. Andronov, E. Leontovich, I. Gordon and A. G. Maier, Qualitative Theory of Second-Order Dynamic Systems, A Halsted Press Book, John Wiley and Sons, [9] C. Chicone, Ordinary Differential Equations with Applications, Springer- Verlag, [1] V. Dubovik, Around Lotka-Volterra Kind equations and nearby Problems, Theor. Popul. Biol, 3 (), [11] E. Canate, W. Fong, C. Severiche, Y. Marrugo and J. Jaimes, Model dynamics of osenzweig - MacArthur considering the proportional refuge function to the number of dams, Contemporary Engineering Sciences, 11 (18), eceived: March 6, 18; Published: April 7, 18